Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs
This work addresses the lack of rigorous error quantification in machine learning for PDE-defined PINNs, providing a method for certified predictions in scientific computing applications.
The authors tackled the problem of quantifying prediction error in physics-informed neural networks (PINNs) by deriving a rigorous a posteriori error bound that can be computed without knowing the true solution, using only a priori information about the underlying PDEs, and demonstrated it on four equations including transport, heat, Navier-Stokes, and Klein-Gordon.
Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.